1.1 Introduction to Quantitative Reasoning
Specific Objectives
Students will understand that



quantitative reasoning is the ability to understand and use quantitative information. It is a
powerful tool in making sense of the world.
media statements often generate more questions than answers.
relatively simple math can help make sense of complex situations.
Students will be able to





identify quantitative information.
rewrite quantitative statements to improve clarity.
round numbers.
name large numbers.
work in groups and participate in discussion using the group norms for the class.
Problem Situation: Does This Information Make Sense?
In this lesson, you will learn how to evaluate information you see often in society. You will start with the
following situation.
You are traveling down the highway and see a billboard with this message:
Every year since 1950,
the number of American children
gunned down has doubled.
(1) You do not see the name of the organization that put up the billboard. What groups might have
wanted to publish this statement? What are some social issues or political ideas that this statement
might support?
The information in this statement is called quantitative. Quantitative information uses concepts about
quantity or number. This can be specific numbers or a pattern based on numerical relationships such as
doubling.
(2) You hear and see statements using quantitative information every day – in advertisements,
pamphlets, billboards, etc. People use these statements as evidence to convince you to do things.
What are some examples of things advertisements try to convince you to do?
You often do not know whether the statements you see are true. You may not be able to locate the
information, but you can start by asking if the statement is reasonable. This means to ask if the
statements make sense. You will be asked if information is “reasonable” throughout this course.
This lesson will help you understand what is meant by this question.
(3) In 1995,1 an article published the statement in the Problem Situation. Do you think this was a
reasonable statement to make in 1995? Discuss with your group.
(4) You only have the information in the statement. Without doing any further research, how can you
decide if the statement is reasonable? Talk with your group about different ways in which you might
answer this question.
(5) In the previous question, you thought about ways to decide if the statement was reasonable. One
approach is to start with a number you think is reasonable for the first year. Put this number into
the table below for the year 1950. Complete the second column of the table by calculating the
number using the statement on the billboard. Do NOT complete the third column right now.
Year
Number of Children
Rounded (using words)
1950
1960
1970
1980
1990
1995
(6) Fill in the third column of the table by rounding the numbers to one or two significant figures.
(7) Does the number you predicted for the number of children shot in 1995 seem reasonable? What
kind of information might help you decide?
1Best,
J. (2001). Damned lies and statistics. University of California Press: Berkeley and Los Angeles.
About This Course
This course is called a quantitative reasoning course. This means that you will learn to use and
understand quantitative information. It will be different from many other math classes you have taken.
You will learn and use mathematical skills connected to situations like the one you discussed in this
lesson. You will talk, read, and write about quantitative information. The lessons will focus on four
themes:




Citizenship: You will learn how to understand information about your society, government, and
world that is important in many decisions you make.
Personal Finance: You will study how to understand and use financial information and how to
use it to make decisions in your life.
Medical Literacy: You will learn how to understand information about health issues and medical
treatments.
Physical World: You will learn how to understand scientific information about how the world
works.
This lesson is part of the Citizenship theme. You learned about ways to decide if information is
reasonable. This can help you form an opinion about an issue.
Today, the goal was to introduce you to the idea of quantitative reasoning. This will help you understand
what to expect from the class. Do not worry if you did not understand all of the math concepts. You will
have more time to work with these ideas throughout the course. You will learn the following things:



You will understand and interpret quantitative information.
You will evaluate quantitative information. Today you did this when you answered if the
statement was reasonable.
You will use quantitative information to make decisions.
Example from January 2016 that shows the need for understanding quantitative reasoning and
large numbers. This has been repeatedly posted on Facebook.
Lesson 1.2 Eighteen Trillion and Counting
Specific Objectives
Students will understand that
• 1 billion = 1,000 x 1,000 x 1,000.
• the representations, one billion, 1,000,000,000, and 109 have the same meaning.
• Scientific notation is a different way to represent large numbers
Students will be able to
• calculate quantities in the billions.
• convert units from inches to feet and feet to miles.
• Compare large numbers
• Write large numbers in scientific notation
Problem Situation 1: How Big Is a Billion or a Trillion?
A large economic and political concern is the federal deficit, the amount of money spent by the federal
government in excess of revenue collected. The federal budget deficit for 2014 was approximately
$483 billion2. The total accumulated federal debt is about $18.1 trillion.3
It is difficult to understand just how big a billion or a trillion is. Here is a way to help you think about it.
1 million = 1,000 x 1,000
= 1,000,000
= 106
1 billion = 1,000 x 1,000 x 1,000
= 1,000,000,000
= 109
1 trillion = 1,000 x 1,000 x 1,000 x 1,000 = 1,000,000,000,000 = 10 12
2
https://www.cbo.gov/publication/44716
3
http://www.treasurydirect.gov/NP/debt/current
(1) The National Debt is shown on the right axis of the above graph.
(a) What is a simpler way to express 2000 billion dollars?
(b) What is a simpler way to express 18000 billion dollars?
Look at website: Visualize the debt
The following questions will give you another way to think about how big these numbers are.
(2) Imagine a stack of 1,000 one‐dollar bills, which is about 4.3 inches tall. Complete the following
steps, and in each write your calculations clearly so that someone else can understand your
work. Note: include units such as ft or mi with your answers.
(a) Imagine combining 1,000 stacks of 1,000 one‐dollar bills. How much money is in the stack?
How tall would that stack be? How tall is the stack measured in feet? (12 inches = 1 foot).
$ _____________________
Height:______________________
(b) Imagine combining 1,000 stacks like the result in Part (a). How much money is in the stack?
How tall is the stack measured in feet? How tall is the stack measured in miles? (5280 feet =
1 mile)
$ _____________________
Height:______________________
(c) Imagine combining 1,000 stacks like the result in Part (b). How much money is in the stack?
How tall is the stack measured in miles?
$ _____________________
Height:______________________
(d) Which of the stacks of money (in a, b, or c) is closest to the federal budget deficit in 2014?
Problem Situation 2: Scientific Notation
You saw that 1 billion can be written as 1,000,000,000 or represented as 109. How would 2 billion be
represented? Since 2 billion is 2 times 1 billion, then 2 billion can be written as 2 x 109. Writing
numbers in this way is called scientific notation. Scientific notation is used primarily for writing very
large numbers and very small numbers. In this lesson, we will focus on large numbers.
The form for scientific notation is M x 10n where 1 ≤ M < 10. The exponent equals the number of
decimal places between the location of the decimal in the number as written in standard notation and
the number written in scientific notation. For example, if the number 500 were to be written in
scientific notation, it could be thought of as 5 x 100 or 5 x 10 x 10 which is equal to 5 x 10 2. The decimal
point has been moved 2 places which corresponds to the exponent of 2.
(3) Write the numbers below in both standard notation and scientific notation.
Standard Notation
Scientific Notation
four hundred
23 thousand
7.2 million
(4) The table below contains the national deficit and national debt for various years. Write the number
in both standard notation and scientific notation.
Standard Notation
Scientific Notation
1965 deficit: $1.41 billion
2009 deficit: $1412.69 billion
2014 deficit: $484.6 billion
2014 debt:
$18.2 trillion
Problem Situation 3: Comparing the Sizes of Numbers
One of the skills you will learn in this course is how to write quantitative information. A writing
principle that you will use throughout the course is given below followed by some questions to
practice the principle.
Writing Principle: Use specific and complete information. The reader should understand what you
are trying to say even if they have not read the question or writing prompt. This includes
•
•
information about context, and
quantitative information.
(5) A headline in 2014 read “Scott vetoes $69 million in $77‐billion state budget”4. Is the $69 million a
small or large portion of the total state budget? Which of the following statements best describes
the relationship?
(a) The portion vetoed is a very small part of the entire state budget
(b) $69 million is about a thousandth of $77 billion
(c) The $69 million vetoed is a very small part of the entire state budget of $77 billion
(d) The $69 million that was vetoed is about one tenth of one percent of the total $77 billion state
budget.
(6) The federal budget in 2012 included $471 billion for Medicare and $47 billion for International
Affairs. Write a statement that compares the two quantities.
(7) The federal budget deficit for 2013 was approximately $680 billion. Write a statement
comparing the federal deficit to one of these:
Median household annual salary in the US: about $60,000
Price of 4 years at Pacific Lutheran University: about $200,000
Cost of a celebrity’s mansion in Beverly Hills: about $4 million
Cost of 787 Dreamliner airplane: about $250 million
Cost of the International Space Station: about $100 billion
4
http://www.tallahassee.com/story/news/2014/06/03/scott‐vetoes‐million‐billion‐state‐budget/9901117/
1.3 Algebra and Scientific Notation with Small Numbers
Specific Objectives
Students will understand that


in algebra, numbers and variables can be combined to produce expressions, equations and
inequalities.
numbers between 0 and 1 can be written using scientific notation. The exponent will be
negative.
Students will be able to




distinguish between expressions, equations and inequalities.
convert between standard and scientific notation.
multiply and divide using scientific notation, both by hand and with a calculator.
apply the rules of exponents to variables.
Lecture
At the root of much of the scientific and technical knowledge that we have is mathematics.
We start with numbers and evolve to algebra.
Numbers (whole, integers, real, fractions, decimals)
Intervals expressed with interval notation (low, high). (0,1) or [1,oo), etc
Patterns: How many squares in each? We can count or find the area by multiplying the length
times the width.
2x2=4
3x2=6
4 x 3 = 12
Length x width = Area
LW = A or A = LW
More patterns:
Algebraic statements: Expressions, Equations, Inequalities
- give examples of each
The emphasis in this class is for students to learn to apply math to real world situations. But the role of
mathematics is not always visible. Rather it is often like the skeleton of a body. We don’t see the
skeleton, but it supports all the other parts of the body that we do see and use. Mathematics underlies
all the science and technology that we use to understand our world and make it progressively better.
Therefore, in this book we will take occasional detours from the application of math to study pure
mathematics to enhance your understanding of the structure upon which much of our understanding of
the world is built.
In the first two lessons, we have worked with numbers, but numbers are only one part of mathematics.
Numbers are initially understood through the study of arithmetic, however arithmetic can be limiting.
These limits can be overcome by finding patterns in the numbers and defining these patterns. This leads
to the concept of variables and the branch of mathematics called algebra. For example, we can use
arithmetic to find out how far we would travel at a given speed based on the number of hours we are
traveling. If the speed of our car is 60 miles per hour and we drive for 1 hour, then we will travel 60
miles. If we travel 2 hour we will travel 120 miles. This is shown using only numbers as:
60 = 60*1
120 = 60 * 2
180 = 60 * 3
There is a pattern that can be observed. The pattern shows that the distance we travel is equal to the
speed of the car times the number of hours at that speed. Explaining it in a sentence takes long, so we
use letters, known as variables, to represent certain phrases so we can shorten it.
Thus, if d represents the distance we travel, r represents the rate or speed of the car, t represents the
time, in hours, then the sentence can be represented as a word equation: 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑟𝑎𝑡𝑒 ∙ 𝑡𝑖𝑚𝑒
and as an algebraic equation: 𝑑 = 𝑟𝑡.
We now have a relationship between variables that is true and that can be applied in a variety of
situations.
One of the challenges of learning math is to recognize patterns in the numbers and variables. Without
seeing the patterns, it is difficult to know which rules apply to which numbers and variables. Therefore,
this lesson will introduce you to some of the big patterns you should learn to recognize.
Numbers make use of 10 basic numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These can be combined to make larger
numbers or written as fractions or decimals to make parts of numbers. Numbers can be classified in a
variety of ways, such as:
Counting numbers {1,2,3, …}
Whole numbers {0, 1, 2, 3, …}
Integers {…-3, -2, -1, 0, 1, 2, 3, …}
Decimals such as 0.4, -3.56,
2 7
Fractions: 3, 4
A note about fractions. If both numbers in a fraction are positive, and if the numerator (number on top)
is smaller than the denominator (number on bottom), the fraction will have a value between 0 and 1. If
the numerator is larger than the denominator, the fraction will have a value greater than 1.
Because it will be useful in a short time, a brief introduction will be made to interval notation. Interval
notation is made with parentheses or brackets enclosing two numbers separated by a comma. The
lowest number is always written first. Thus it will look like (low,high) or [low,high]. For example,
numbers “between 0 and 1” can be written is (0,1). The use of parentheses indicates we will not include
a 0 or a 1. If we wanted to include both a 0 and a 1 we would use brackets [0,1]. To indicate numbers
greater than 1 using interval notation we incorporate the infinity symbol, ∞. If we don’t want to include
1, this range of numbers is written as (1, ∞). If we want to include 1, it is written [1, ∞). A parentheses
is always used next to the infinity symbol.
Real numbers are all numbers between negative infinity and positive infinity, (-∞,∞).
Variables are letters used to represent an unknown value. Typically letters from the English alphabet
are used with x, y, z, a, b, c being the most common. At times, letters from the Greek alphabet are also
used. Examples include µ (mu) and σ (sigma).
In the study of algebra, we can create combinations of numbers, variables and operation symbols.
These combinations can be categorized as expressions, equations, and inequalities.
Expressions: 𝑥,
3𝑦 (meaning 3 times y), 𝑏 + 3 (meaning 3 is added to b), 𝑥 2 (meaning 𝑥 ∙ 𝑥)
Equations are used when 2 expressions are equivalent. Equations contain an equal sign (=). Examples
include: 𝑥 + 2 = 2𝑥 − 5 and 𝑦 = 4𝑥 + 3.
Inequalities are used when one expression is more or less than another expression. Inequalities contain
an inequality sign (<, >, ≤, ≥). Examples include: 𝑥 + 2 > 2𝑥 − 5 and 𝑦 < 4𝑥 + 3.
1. Identify each of the following as an expression, equation, or inequality.
8𝑥 = 24
Expression
Equation
Inequality
4𝑥 + 5 < 2(𝑥 − 3)
Expression
Equation
Inequality
5(2𝑥 + 9) = 3𝑥 − 7
Expression
Equation
Inequality
4𝑥 − 6 + 7𝑥 − 3
Expression
Equation
Inequality
5(3𝑦 + 2)
Expression
Equation
Inequality
3𝑥 + 7𝑦 = 14
Expression
Equation
Inequality
One of the expressions given above is x2. This is read as “x squared” or “x to the second power.” The 2
is the exponent. The x is the base. An exponent tells how many times to use the base in a
multiplication. For example, x2 = xx, x3 = xxx. This relates to the scientific notation you learned in the
last lesson. For example, 1000 can be written as 1 x 103 where 103 = 10 x 10 x 10.
Scientific notation will be used to introduce us to the rules of exponents. The left column below
contains numbers that start at 1 million and decrease by a factor of 10. That is, successive numbers are
1/10 the size of the prior number. In the column at the right is the same number recorded in scientific
notation. The cells that are shaded have a line where the exponent should be.
2. Find the pattern in the numbers then put the appropriate exponent on the line.
1,000,000
100,000
10,000
1,000
100
10
1
0.1
0.01
0.001
0.0001
1 x 106
1 x 105
1 x 104
1 x 103
1 x 102
1 x 101
1 x 10__
1 x 10__
1 x 10__
1 x 10__
1 x 10__
Verify your answers are correct before continuing.
Answer the following questions to help you develop a big-picture view of scientific notation.
3. For numbers that are greater than 1, is the power of 10 a positive or negative number? ___________
4. For numbers that are between 0 and 1, is the power of 10 a positive or negative number? _________
5. What is the power of 10 that produces a 1?
_____________
We will now investigate the numbers in the interval (0,1) to understand the meaning of scientific
notation when the exponents of 10 are negative. You have learned the following in prior math classes.
1
0.1 is one-tenth, which is written 10.
0.01 is one-hundredth, which is written
1
100
but which could be written as
1
1
.
102
1
0.001 is one-thousandth, which is written 1000 but which could be written as 103.
6. If 10-1 =
1
,
10
and 10-2=
1
,
102
and 10-3 =
1
,
103
then what would you conclude about 10-4? _____________
7. What do you conclude about 10-n? ____________________
The reasoning behind scientific notation for small numbers (0,1)
The diameter of a human egg cell is 130 µm (micrometers). Since a micrometer is 0.000001 meters in
length, then 130 micrometers could be written as 0.000130. This can also be written as
factor of 100 is removed from each number, this could be written as
1.30
,
104
1.30
10,000
130
.
1,000,000
If a
which can then be written as
which, based on what was shown with negative exponents, can be written as 1.30 x 10-4.
The quicker way to convert from standard notation to scientific notation is to count the number of
decimal places between the location of the decimal point and the desired location of the decimal point
(with exactly one digit to the left). If the original number is between 0 and 1, the exponent in scientific
notation will be negative. If the original number is greater than 1, the exponent will be positive.
Standard Notation
120
120,000
120,000,000
(120 million)
Scientific Notation
1.2 x 102
1.2 x 105
1.2 x 108
Standard Notation
0.0120
0.0000120
0.000120
(120 millionth)
Scientific Notation
1.2 x 10-2
1.2 x 10-5
1.2 x 10-4
Scientific notation – big and small numbers.
While scientific notation can be used for such things as money and populations, it is more typically used
for science. Things such as the mass of planets or the distance between bodies in the solar system
benefit from the use of scientific notation. For such large numbers, the exponent for 10 will be positive.
But science also looks at very small objects, such as cells and molecules and atoms. For small objects,
the exponent of 10 will be negative.
8. To appreciate the range of sizes that scientists study, match each length in the first table with the
appropriate number of meters in the second table.
Distance from earth to the
moon
Distance from earth to Mars (at
their closest point)
Distance from earth to the sun
Kitchen counter height from
floor, baseball bat, walking stick
Size of a water molecule or
carbon atom
Size of a virus
Length of a grain of rice
Distance from earth to Pluto (at
their closest point)
Size of a chromosome or red
blood cell
One trip around Lake Waughop
Distance from earth to space
station
Powers of 10
Units of meters
Measurement (from the list above) Change font color below to see answers
1012
1011
1010
108
105
103
100
10-3
10-6
10-8
10-10
Distance from earth to Pluto (at their closest point)
Distance from earth to the sun
Distance from earth to Mars (at their closest point)
Distance from earth to the moon
Distance from earth to space station
One trip around Lake Waughop
Kitchen counter height from floor, baseball bat, walking stick
Length of a grain of rice
Size of a chromosome or red blood cell
Size of a virus
Size of a water molecule or carbon atom
The form for numbers written in scientific notation is M x 10n where 1≤M<10 and n is an integer. The
upper case M is called the mantissa. We will call the numbers that look like they are written in scientific
notation but that have a mantissa outside of the interval [1,10) pseudo-scientific notation. Thus, 25 x
103 is written in pseudo-scientific notation and should be rewritten as 2.5 x 104 to be in proper scientific
notation.
9. Identify if the number is written correctly in scientific notation or if it is written in pseudo-scientific
notation.
3.4 x 108
Scientific Notation
pseudo-scientific notation
62.3 x 104
Scientific Notation
pseudo-scientific notation
1.578 x 10-3
Scientific Notation
pseudo-scientific notation
0.45 x 10-6
Scientific Notation
pseudo-scientific notation
5.667 x 103
Scientific Notation
pseudo-scientific notation
There are a variety of skills that students should be able to do with scientific notation and exponent
rules.








Convert from standard notation to scientific notation.
Convert from scientific notation to standard notation.
Convert from pseudo-scientific notation to scientific notation.
Multiply scientific notation numbers by hand and with a calculator
Multiply variables with exponents
Divide scientific notation numbers by hand and with a calculator
Divide variables with exponents
Raise variables with exponents to a power.
Convert from standard notation to scientific notation.
a. Identify if the number in standard notation is in the interval (0,1) or (1, ∞).
b. Count the number of places the decimal must be moved so there is only one (non-zero) digit
to the left of it.
c. Write the mantissa followed by “x 10”. The power of 10 will be negative if the original
number is in the interval (0,1) and positive if the original number is greater than 1. The
exponent will be equal to the number of places the decimal was moved.
Example: Write 0.0045 in scientific notation.
a. This number is in the interval (0,1). Therefore we know the exponent will be negative.
b. From 0.0045 to 4.5, the decimal point must be moved 3 places.
c. 4.5 x 10-3
Example: Write 4,500 in scientific notation.
a. This number is in the interval (1, ∞). Therefore we know the exponent will be positive.
b. From 4,500 to 4.5, the decimal point must be moved 3 places.
c. 4.5 x 103
Convert from scientific notation to standard notation.
a. Identify if the exponent is positive or negative so you will know if you are writing a number in
the interval [1, ∞) or (0,1).
b. Move the decimal point the number of places indicated by the exponent
Example: Write 3.7 x 106 in standard notation.
a. the exponent 6 is positive, so this will be a big number (1, ∞).
b. The decimal point will be moved 6 places from its current position in 3.7 to its new position in
3,700,000.
Example: Write 3.7 x 10-6 in standard notation.
a. The exponent 6 is negative, so this will be a small number (0,1).
b. The decimal point will be moved 6 places from its current position in 3.7 to its new position in
0.0000037. (notice that there are only 5 zeros).
Convert from pseudo-scientific notation to scientific notation.
The population of the US is about 320 million. This could be written as 320 x 106 but because this is not
true scientific notation since the mantissa has more than one digit to the left of the decimal, it is called
pseudo-scientific notation.
a. write the mantissa in scientific notation
b. add the two exponents
Example: Write 320 x 106 in scientific notation
a. (3.20 x 102) x 106
b. 3.20 x 108
Example: An E. Coli Bacterium has a length of 0.6 micrometers. Write this number in scientific notation.
a. (6 x 10-1) x 10-6
b. 6 x 10-7
Example: A virus has a length of 125 nm (nanometers). This number in pseudo-scientific notation is
125 x 10-9. Write this number in scientific notation.
a. (1.25 x 102) x 10-9
b. 1.25 x 10-7
Multiply scientific notation numbers by hand and with a calculator
How would we multiply 1 x 102 times 1 x 103? We know that 102 = 100 and 103 = 1000. Therefore we
should expect to get the same result as 100 x 1000 = 100,000. We also know that 102 = 10x10 103 =
10x10x10 Therefore, 102 x 103 = 10 x 10 x 10 x 10 x 10 = 100,000 = 105. If we were to generalize from
this one example (not always a good idea), we could see that by adding the exponents of 10 2 and 103 we
get 105.
10. What is (1 x 105) · (1 x 106)? _____________________
11. Let’s make it a little more complicated. What is (2 x 105) · (3 x 106)? In this case, multiply the
mantissas and then multiply the powers of 10 by adding the exponents.
(2 x 105) · (3 x 106) = __________________________
12. What is (4 x 105) · (5 x 106)? In this case, your answer will be in pseudo-scientific notation and must
be converted into scientific notation.
(4 x 105) · (5 x 106) = ____________________________ = ___________________________
13. What is (6 x 10-5) · (4 x 103)?
Scientific and graphing calculators have the ability to make calculations using scientific notation. If your
calculator is made by Texas Instruments, find the EE key. EE means “Enter Exponent”. It may be a
second function. If it is made by Sharp or Casio, look for the EXP key.
Example: Multiply (6.3 x 1012) · (7.8 x 1015) using your calculator.
Enter 6.3 then 2nd EE 12 (pay attention to how your calculator shows the scientific notation).
Press the X key
Enter 7.8 then 2nd EE 15
Press the Enter key
Your answer should be 4.914E28 which is equal to 4.914 x 1028.
Multiplying variables with exponents
We just saw that 102 · 103 = 105. Since the base remains the same and the exponents are added, then
we can generalize this using variables. For example, x2 · x3 = x5, or more generally,
The rule is
xa · xb = xa+b
When the bases are the same, the product is found by adding the exponents.
14.
a. Multiply x5x6 _____________
b. Multiply x-4x-6 _____________
c. Multiply x-12x8 ____________
Dividing scientific notation numbers by hand and with a calculator
How would you divide 1 x 105 by 1 x 103?
These could be written as
1×105
1×103
=
10×10×10×10×10
10×10×10
= 102
Another way to do the division is by subtracting the exponent in the denominator from the exponent in
the numerator (5 – 3 = 2).
Example: What is 1 x 103 divided by 1 x 105?
These can be written as
1×103
1×105
=
10×10×10
10×10×10×10×10
=
1
102
However if we subtract the exponent 5 from the exponent 3 we get 10-2. From this we can see
1
that 10−2 = 102.
1
We can generalize this to say that 10−𝑛 = 10𝑛. Therefore, a negative exponent can be written as a
positive exponent by moving the term to the other side of the division bar.
15. What is (6 x 107) / (3 x 104)? In this case, divide the mantissa in the numerator by the mantissa in
the denominator then subtract the exponent in the denominator from the exponent in the
numerator to find the power of 10.
(6 x 107) / (3 x 104) = __________________
16. What is (4 x 10-6) / (8 x 109)? This will give you a pseudo-scientific notation number. Convert it to
proper scientific notation.
(4 x 10-6) / (8 x 109) = _________________________ = ____________________________
17. Use your calculator to find (6.3 x 1012) / (7.8 x 1015).
___________________________.
Dividing variables with exponents
There are 3 situations we encounter when dividing terms with exponents. Assuming the base is the
same, then the exponent in the numerator could be larger, the same as, or smaller than the exponent in
the denominator. The process is the same for all three methods.
Divide
Divide
Divide
𝑥5
𝑥5
𝑥3
𝑥3
𝑥3
𝑥3
𝑥3
𝑥3
𝑥3
𝑥3
𝑥5
𝑥5
=
=
=
𝑥𝑥𝑥𝑥𝑥
𝑥𝑥𝑥
𝑥𝑥𝑥
𝑥𝑥𝑥
= 𝑥𝑥 = 𝑥 2
=1
𝑥𝑥𝑥
𝑥𝑥𝑥𝑥𝑥
=
1
𝑥𝑥
=
1
𝑥2
= 𝑥 −2
The rule is
18.
a. Divide
c. Divide
𝑥8
𝑥3
𝑥7
𝑥 −4
𝑥𝑎
𝑥𝑏
= 𝑥 𝑎−𝑏
____________
b. Divide
_____________
d. Divide
𝑥 −7
𝑥 −5
𝑥 −4
𝑥 −4
____________
______________
Raise variables with exponents to a power.
The final exponent rule to learn is how to raise an exponent to a power. An example is (𝑥 2 )3 .
Since an exponent applies to the base it is next to, then the power of three indicates to multiply
𝑥 2 𝑥 2 𝑥 2 = 𝑥 6 . The quicker way is to multiply the exponent 3 times the exponent 2 to get the 6.
The rule is (𝑥 𝑎 )𝑏 = 𝑥 𝑎𝑏
19. Raise x-3 to the 5th power.
(𝑥 −3 )5 = _________________
1.4 Water Footprint
Specific Objectives
Students will understand that



the magnitude of large numbers is seen in place value and in scientific notation.
proportions are one way to compare numbers of varying magnitudes.
different comparisons may be needed to accurately compare two or more quantities.
Students will be able to





express numbers in scientific notation.
estimate ratios of large numbers.
calculate ratios of large numbers.
use multiple computations to compare quantities.
compare and rank numbers including those of different magnitudes (millions, billions).
Numerous scientists have conjectured about how long humans can sustain ourselves, as we cruise the
solar system in our self-contained environment. One of the most important natural resources that
humans need for survival is water. An influential United Nations report published in 2003 predicted that
severe water shortages will affect 4 billion people by 2050. This report also said that 40 percent of the
world’s population did not have access to adequate sanitation facilities in 2003.5 Humans need clean
water not just for drinking, but for necessary tasks such as sanitation, growing food, and producing
goods.
You will use a measure of water consumption called a “water footprint” that includes all of the ways
that people use fresh water. According to Waterwiki.net, “The water footprint of an individual, business,
or nation is defined as the total volume of freshwater that is used to produce the goods and services
consumed by the individual, business, or nation.”6 Goods are physical products such as food, clothes,
books, or cars. Services are types of work done by other people. Examples of services are having your
hair cut, having a mechanic fix your car, or having someone provide day care for your children. Fresh
water is often used to make goods and to provide you with services.
In this lesson, you are going to compare the populations of China, the United States, and India. You will
go on to look at the water footprint for each nation as a whole and per person (“per capita”) to make
some comparisons and to consider what this information might mean for the planet’s sustainability—
that is, Earth’s ability to continue to support human life. While there is no one definition of
sustainability, most agree that “sustainability is improving the quality of human life while living within
the carrying capacity of supporting eco-systems.” Carrying capacity refers to how many living plants,
animals, and people Earth can support into the future.
5
Retrieved from Rajan, A. Forget carbon: you should be checking your water footprint. Monday, 21 April 2008. Link
[http://www.independent.co.uk/environment/green-living/forget-carbon-you-should-be-checking-your-water-footprint812653.html]
6
Retrieved from http://waterwiki.net/index.php/Water_footprint
Problem Situation 1: Comparing Populations
You will begin by thinking of various ways you can compare different countries’ populations. Scientific
notation will be a useful tool because it is a way to write large numbers. Recall that a number in
scientific notation is written in the form: M x 10n where 1 ≤ M < 10; and n is an integer.
(1) In 2014, the population of the United States was 317,000,000. Earth’s population was about
7.2 billion. Write these numbers in scientific notation.
(2) (a) Write a ratio comparing the U.S. population to the world’s population using a fraction. How
could you simplify the fraction so that it has smaller numbers in both the numerator and
denominator?
(b) The ratio in part (a) compared a part of the world’s population to the whole world’s population,
so it is reasonable to re-write it as a percentage. Calculate the percentage Write a contextual
sentence for this result, which is a sentence that follows the Writing Principle (That is: Use specific
and complete information. Someone who reads what you wrote should understand what you are
trying to say, even if they have not read the question or writing prompt.)
(3) In 2014, the population of China was 1.39 billion. Compare China’s 2014 population to the world
population with a ratio. Write your answer as a percent and as a fraction. Consider how many
decimals to give in your final answer.
(4) Compare China’s population with the population of the United States using a ratio with the U.S.
population as the reference value (denominator). Write a sentence that interprets this ratio in the
given context (follow the Writing Principle).
Problem Situation 2: Comparing Water Footprints
The population of the United States is the third largest in the world, exceeded only by China and India.
Which country uses the most water? Which country uses the most water per person? We’ll now
explore this.
The table below gives the population and water footprints of China, India, and the United States for
2011. Notice the units used, given in the column labels.
Population7
Total Water Footprint8
(in millions)
(in 109 cubic meters per year)
China
1,346
1,368
India
1,241
1,144
312
821
Country
United States
(5) Re-write the information from the table above into the table below, expressing the numbers in
standard notation and in scientific notation, rather than with the units used above. The cell for
China’s population is filled in already. Complete the other five cells.
Country
Population
Total Water Footprint
(in cubic meters per year)
1,346,000,000
China
1.346 x 109
India
United States
7
8
http://www.prb.org/pdf11/2011population-data-sheet_eng.pdf
http://www.pnas.org/content/109/9/3232.abstract
(6) (a) Complete this contextual sentence:
The United States had about _______________ million people in 2011.
(b) Write a contextual sentence describing the United States water footprint, using a number and
word combination as in part (a).
(7) Notice that the countries are listed in the table from highest to lowest population. Using the data on
Total Water Footprint, rank the countries (from highest to lowest) according to their total water
footprint.
(8) In the fourth column in the table we want to put the amount of water used on average by each
person in the country. This is the “Water footprint per person” (which is the same as “water
footprint per capita”) for each country.
- Label this topic in the top row of column four, and write the units.
- Calculate and write in the table the water footprint per person for each country.
(9) Rank the countries in order of water footprint per person (“per capita”) from highest to lowest.
(10) In 2011 the average person in the United States used how many times more water than the average
person in China? Justify your response.
(11) Write a contextual sentence to explain the meaning of your answer to the previous question.
(Remember the Writing Principle: Use specific and complete information. Someone who reads what
you wrote should understand what you are trying to say, even if they have not read the question or
writing prompt.)
(12) In 2011 how many times larger was the population of China compared to the population of the U.S.
Justify your response.
(13) Write a few sentences relating the information of #11 and 12 and what this might mean in terms
of sustainability.
1.5 Dimensional Analysis
Specific Objectives
Students will understand that


units provide meaning to numbers in a context.
the units in an expression may be used as a guide for needed conversions.
Students will be able to


use units to determine which conversion factors are needed for dimensional analysis.
use dimensional analysis to convert units and rates.
Introduction to Dimensional Analysis
Most numbers used in the real world have units attached, which clarify what the number is referring to.
Examples of units are gallons, dollars, meters, miles, and pounds. Some units are for geometric
measurements such as area or volume. Many disciplines such as medicine or engineering have special
units for use in their field.
This lesson will focus on a valuable strategy for converting from one set of units to another. This skill is
called dimensional analysis. It is also known as unit analysis, unit-factor conversion, and the factorlabel method.
The dimensional analysis strategy is based on three familiar ideas:


A fraction with equivalent expressions in the numerator and the denominator is equal to the
7 𝑓𝑡 3
3a
25
number one. Examples:
=1
=1
=1
7 𝑓𝑡 3
3a
7
Multiplying something by the number 1 does not change its value.
Example:

$4.99
$4.99
1 =
2 pounds
2 pounds
However, multiplying by the number 1 can be used to change the way a something appears.
4
4 2
8
Example: to rewrite fifths as tenths:
=
=

5
5 2 10
When multiplying fractions, if a factor occurs in both the numerator and the denominator, it can
be divided out. The common factor may be a number or a variable. Example:
3g m 3  g  m
3 g  m 3 g m
3 g
3 g
3g
 =
 =
1 =
=
=
=
7m 2
7m2
72m 72 m
72
72
14
The key to unit conversions with dimensional analysis is multiplying by the number one in the form of a
conversion fraction. Conversion fractions are fractions with different units in the numerator and
denominator but in which the value of the numerator equals the value of the denominator. Examples:
Since 3 feet is equal to 1 yard, the fraction
3 feet
1 yard
= 1 and also the fraction
= 1.
1 yard
3 feet
Since 1 hour equals 60 minutes, the fraction
1 hour
60 minutes
= 1 and also the fraction
= 1.
60 minutes
1 hour
To use the dimensional analysis method:
Start with the original quantity and multiply it by the number 1 written as a conversion fraction of two
units so that the units you don’t want can divide out of the numerator and denominator.
Example 1. Convert 300 feet to yards.
● Start with 300 feet
● Multiply by a conversion fraction with feet in the denominator (so that the “feet” divides out of
both numerator and denominator), and in the numerator you want to have yards
 1 yard 
 1 yard 
  100 yards
300 feet 
 = 300 feet
 3 feet 
 3 feet 
Example 2. Convert 30 ounces of weight to the metric unit of grams. Determine conversion factors
from the Unit Equivalencies table below, in the part for weight and mass.
● Start with 30 ounces.
● Multiply by a conversion fraction with ounces in the denominator, and you’d like grams in the
numerator. However, in the equivalencies table, it doesn’t say how many ounces equals how many
grams. But it does give equivalencies for ounces to pounds and then pounds to grams. So we use
two conversion fractions.
30 ounces 1 pound 453.6 grams 30 ounces 1 pound 453.6 grams




=
=
1
16 ounces
1 pound
1
16 ounces
1 pound
30  453.6 grams
16
= 850.5 grams
Here is a link to a video that gives more examples of converting units using dimensional analysis.
https://www.youtube.com/watch?v=7N0lRJLwpPI
To create conversion fractions equal to 1, you must know which units are equivalent. The table below
provides some unit equivalencies.
Notice the structure of the table:



Units of the U.S. system are in the left column. Units of the Metric system are in the right column.
The middle column shows some equivalencies between U.S. and metric units.
Different types of measurements are in different rows: Length, Area, Capacity or Volume, Weight or
Mass.
When you want to find what a particular unit is equivalent to, you need to locate the unit in the
correct row and column of the table.
USCS (US Customary
System)
Unit Equivalencies
USCS – Metric
Metric or SI
Length
12 inches (in) = 1 foot (ft)
3 feet (ft) = 1 yard (yd)
1760 yards(yd) = 1 mile (mi)
5280 feet (ft) = 1 mile (mi)
2
1 square mile (mi ) = 640 acre (ac)
1 acre (ac) = 43,560 square feet
(ft2)
1 inch (in) = 2.54 centimeters(cm)
0.62 miles (mi) = 1 kilometer (km)
Area
2.471 acre (ac) = 1 hectare (ha)
1 square mile (mi2) = 2.59 square
kilometers (km2)
1000 millimeters (mm) = 1 meter (m)
1000 meters (m) = 1 kilometer (km)
100 centimeters (cm) = 1 meter (m)
1 square kilometer (km2) = 100
hectare (ha)
1 hectare (ha) = 10,000 square
meters (m2)
Volume
8 ounces (oz) = 1 cup (c)
2 cups (c) = 1 pint (pt)
2 pints (pt) = 1 quart (qt)
4 quarts (qt) = 1 gallon (gal)
1 cubic foot (ft3)=7.481 gallons (gal)
1 quart (qt) = 0.946 liters (L)
1000 milliliters (ml) = 1 liter (L)
1000 liters (L) = 1 cubic meter (m3)
Weight or Mass
16 ounces (oz) = 1 pound (lb)
2000 pounds (lb) = 1 ton
2.20 pounds (lb) = 1 kilogram (kg)
1 pound (lb) = 453.6 grams (g)
1000 milligrams (mg) = 1 gram (g)
1000 grams (g) = 1 kilogram (kg)
1000 kilograms = 1 metric ton
Note about rates
A ratio (that is, a fraction) that includes a unit in the numerator that is different from the unit in its
denominator is typically called a “rate”. Rates show how one variable changes for each change in the
second variable. For example, a rate of speed is
35 𝑚𝑖
ℎ𝑟
which can be read as 35 miles per hour.
The word “per” is the fraction line, the division bar. “Miles per hour” means miles divided by 1 hour.
Some common rates are abbreviated:
“miles per hour” is often written as “mph”, and “miles per gallon” is often written as “mpg”.
When doing calculations with a rate, it should be written as a fraction.
Rates can be converted from one set of units to another. This typically requires the use of several
conversion fractions multiplied in sequence.
Example 3: Converting a rate. Flow rate can be measured in cubic meters per hour. If a river
flows at 200 cubic meters per hour, what is the flow rate in gallons per second?
200m3
 Start with 200 cubic meters per hour which can be written as
hour

Then multiply by conversion fractions so that units you don’t want in the end will cancel. To
form the conversion fractions find equivalent units for capacity or volume from the table.
200 𝑚3 1000 𝐿
1 𝑞𝑡
1 𝑔𝑎𝑙
1 ℎ𝑟
1 𝑚𝑖𝑛
𝑔𝑎𝑙
(
)(
)(
)(
)(
) = 14.68
3
ℎ𝑜𝑢𝑟
1𝑚
0.946 𝐿 4 𝑞𝑡 60 𝑚𝑖𝑛 60 𝑠𝑒𝑐
𝑠𝑒𝑐
Multiply numbers in the numerator, divide by numbers in the denominator. In your calculator enter 200
x 1000 / 0.946 / 4 / 60 / 60 enter.
Notice the 200 is not cubed (it is the unit m that is cubed).
Problem Situation 1: Using Dimensional Analysis
(1) In the US we measure height in feet and inches. In other parts of the world height is measured in
meters. Use dimensional analysis convert the height of a person who is 5 feet, 10 inches (70 inches)
into meters.
(2) In the US we measure weight in pounds. In other parts of the world weight is measured in
kilograms. Use dimensional analysis convert the weight of a person who is 180 pounds into
kilograms.
Texting while driving: Washington state law*: “a person operating a moving noncommercial motor
vehicle who, by means of an electronic wireless communications device, sends, reads, or writes a text
message, is guilty of a traffic infraction.”9
Is texting while driving actually a problem? A person might spend only 4 seconds to answer a text. How
far would the car go in that time? It depends on the car’s speed.
(3) Suppose a car is traveling at 35 miles per hour.
(a) Before calculating it, what do you think is the distance the car will travel in 4 seconds?
(b) Use dimensional analysis to calculate the distance.
9
there are exceptions related to emergencies, emergency vehicles, and totally voice-operated systems. From
http://apps.leg.wa.gov/rcw/default.aspx?cite=46.61.668 on 07/17/15
(4) If a car is traveling at 45 miles per hour, how far will it travel in 4 seconds?
(5) If you are driving in Canada, following the posted speed limit of 80 km/hr, how many feet would you
go during the 4 seconds spent texting?
(6) (a) If the typical texting response time is between 2 seconds and 6 seconds, and a car is travelling at
35 mph what is the typical distance a car would travel during a typical texting response?
(b) Express the answer using inequality symbols (such as < or >).
Using Dimensional Analysis for Area and Volume units
See the “Resource: Length, Area, and Volume” page of this text to review the concepts and units of
length, area, and volume.
There are two methods of converting area and volume units. The examples below show both of the
methods. Sometimes one method may be easier than the other.
Example 4A: Area units conversion: Convert 4 square feet to square inches.
Method: Create a conversion fraction with equivalent square units
• First write the question using symbols for the units: Convert 4 ft2 to in2.
• As with other unit conversions, we want to multiply 4 ft2 by a conversion fraction with ft2 in
the denominator and in2 in the numerator. So we need to know how many ft2 equal how
many in2. You may not know this fact, which is fine because we can find it out from knowing
the linear measurement equivalency that 1 ft = 12 in. We will use that to find the area unit
equivalency by squaring each side of the equation:
1 ft = 12 in
(1 ft) 2 = (12 in) 2
1 ft)(1 ft) = (12 in)(12 in)
1•1•ft•ft = 12•12•in•in
1 ft2 = 144 in2
These diagrams giving the geometric view of the algebra. Each of these is one square foot. Find
the area by taking length times width.
1 ft

12 in
1 ft
12 in
Area = (1 ft)(1 ft)
Area = (12 in)(12 in)
= 1•1•ft•ft
= 12•12•in•in
= 1 ft2
= 144 in2
Now we know the area unit equivalency that 1 ft2 = 144 in2. We use that to create the
conversion fraction and complete the unit conversion.
4 𝑓𝑡 2

Conclude:
144 𝑖𝑛2
1 𝑓𝑡 2
= (4•144) 𝑖𝑛2
= 576 𝑖𝑛2
4 𝑓𝑡 2 = 576 𝑖𝑛2
Example 4B: Area units conversion: Convert 4 square feet to square inches.
Method: Use conversion factor of equivalent linear units, and square that conversion factor
• First write the question using symbols for the units: Convert 4 ft2 to in2.
• We want to multiply 4 ft2 by a conversion fraction with ft2 in the denominator and in2 in the
numerator. If we could find a conversion factor for feet in the denominator and inches in the
numerator, then we could square that entire fraction so we’d end up with square units. The
linear measurement equivalency that 1 ft = 12 in. So we’ll use square the fraction (12 in/1ft).
12 𝑖𝑛 2

4 𝑓𝑡 2 ( 1 𝑓𝑡 )

Conclude:
= (4•122) 𝑖𝑛2
= 576 𝑖𝑛2
4 𝑓𝑡 2 = 576 𝑖𝑛2
Example 5A: Volume units conversion: Convert 3,000,000 cubic centimeter to cubic meters.
Method: Create a conversion fraction with equivalent cubic units
•
First re-write in symbols: convert 3,000,000 𝑐𝑚3 to 𝑚3 .
• To figure out the equivalency of the cubic units (how many 𝑐𝑚3 equals how many 𝑚3 ) start
with what we know about the linear units: 100 cm = 1 m. Then cube both values:
100 cm = 1 m
(100 𝑐𝑚)3 = (1 𝑚)3
100•100•100•cm•cm•cm = 1•1•1•m•m•m
1,000,000 𝑐𝑚3 = 1 𝑚3

Use this to equivalency to create the conversion fraction and carry out the conversion.
3,000,000 𝑐𝑚3 •

1 𝑚3
1,000,000 𝑐𝑚3
Conclude: 3,000,000 𝑐𝑚3 = 3 𝑚3
=
3,000,000
1,000,000
𝑚3 = 3 𝑚3
Example 5B: Volume units conversion: Convert 3,000,000 cubic centimeter to cubic meters.
Method: Use a conversion factor of equivalent linear units, and cube that conversion factor
• First re-write in symbols: convert 3,000,000 𝑐𝑚3 to 𝑚3 .
• Find a conversion fraction for cm in the denominator and m in the numerator, and then cube
that fraction. We know 1 m = 100 cm, so use the conversion fraction (1m / 100 cm) and cube
it.
3
1𝑚
3,000,000•∙1•1•1
100•100•100
•
3,000,000 𝑐𝑚3 • (100𝑐𝑚)

Conclude: 3,000,000 𝑐𝑚3 = 3 𝑚3
=
𝑚3 = 3 𝑚3
Example 6: Convert 100 km2 to square miles using linear equivalencies
100 𝑘𝑚2 (
=
1000 𝑚 2
1 𝑘𝑚
100•10002 •1002
2.542 •122 • 52802
) (
100 𝑐𝑚 2
1𝑚
) (
1 𝑖𝑛
2.54 𝑐𝑚
2
) (
1 𝑓𝑡
12 𝑖𝑛
2
) (
1 𝑚𝑖
5280 𝑓𝑡
2
) =
= 38.61 mi2
Here is a link to a video showing more examples of converting area and cubic units:
https://www.youtube.com/watch?v=aFAk8JA4-d8
Problem Situation 2: Length, Area, Volume
We can think of our physical world as having 3 dimensions. A length or distance is one dimension. An
area is 2 dimensions and a volume is 3 dimensions. Examples of the type of units used for each of these
are shown in the table of unit equivalencies.
In this problem, you will determine various dimensions of your classroom.
(7) Before measurements and calculations are done, make a guess for each of the following in the units
provided.
Length of room ___________ feet
____________ meters
___________ ft2
____________ m2
Volume of room ___________ ft3
____________ m3
Area of floor
(8) Measure and record the following information about the classroom.
a) Length of this room in meters: ________________
b) Width of this room in meters: ________________
c) Height of this room in feet: ________________
d) Calculate the area of the floor of this room in square meters: ________________
(9) Length: Use dimensional analysis and the information above to determine these lengths, showing
work.
a) Length of this room in feet:
b) Height of this room in meters:
(10)Area: Above you calculated the area of the floor of the room in square meters.
Convert that area to square feet using dimensional analysis.
Note: There are at least two dimensional analysis ways of doing this. One starts with conversion
fractions in linear units. The other uses conversion fractions for area units.
Write out both of them.
a)
b)
(11)Volume: Find the volume of this room in cubic feet.
(12)Volume: Use dimensional analysis to convert into cubic meters the volume that you just calculated
in cubic feet.
Note: There are at least two dimensional analysis ways of doing this. One starts with conversion
fractions in linear units. The other uses conversion fractions for volume units.
Write out both of them.
a)
b)
(13) Rate: We breathe about 10 liters of air each minute while being fairly still.10 How many cubic feet
of air do we breathe each hour?
10
http://www.arb.ca.gov/research/resnotes/notes/94-11.htm
1.6 The Cost of Driving
Specific Objectives
Students will understand that


units can be used as a guide to the operations required in the problem—that is, factors are
positioned so that the appropriate units cancel.
the units required in a solution may be used as a guide to the operations required.
Students will be able to



write a rate as a fraction.
use dimensional analysis to determine the factors in a series of operations to obtain solutions.
solve a complex problem with multiple pieces of information and steps.
Problem Situation: Cost of Driving
Jenna’s job requires her to travel. She owns a 2006 Toyota 4Runner, but she also has the option to rent
a car for her travel. In either case, her employer will reimburse her the same amount for the mileage
driven using the rate set by the Internal Revenue Service. In 2015, that rate was 57.5 cents/mile. In this
lesson you will explore the question of whether it would be better for Jenna to drive her own car or to
rent a car.
(1) What do you need to know to calculate the cost of Jenna driving her own car?
(2) What do you need to know to calculate the cost of Jenna renting a car?
(3) Gas mileage is rated for either city driving or highway driving. Most of Jenna’s travel will take place
on the highway. For her next trip she needs to drive 193 miles from home, and then return. The
price of gas is $3.50/gallon. Her 4Runner gets 22 miles/gallon. If Jenna rents, she can request a
small, fuel-efficient car such as the Hyundai Elantra, which gets 40 miles/gallon.
(a) For which vehicle would the gas for this trip cost more?
Estimate about how much more it would cost. Explain your strategy.
(b) Calculate what the cost of the gas for the trip would be for each vehicle.
Show your strategy/method.
(4) Using the information below, calculate Jenna’s total cost of driving a rental car for a round trip.




11
Price of gas: $3.50/gallon
Length of trip (one way): 193 miles
Gas mileage of rental car: 40 miles/gallon11
Price of renting the car: $98.98 plus 15.3% tax (Gas is not included in the rental price. The car
starts with a full tank of gas when rented and must be returned to the rental agency with a full
tank)
Retrieved from www.fueleconomy.gov
(5) Using the information below and knowing Jenna already owns the car, calculate the extra cost for
Jenna of driving her own car for this round trip.




Price of gas: $3.50 per gallon
Length of trip (one way): 193 miles
Gas mileage of Jenna’s car: 22 miles per gallon12
Insurance, registration, taxes: Jenna spends $2,000 a year on these expenses and last year
she drove about 21,600 miles (Note that these costs can vary greatly depending on location
and the individual.)
Maintenance costs for Jenna’s car:
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General maintenance (oil and fluid changes): $40 every 3,000 miles
Tires: Tires for Jenna’s car cost $920; they are supposed to be replaced every 50,000 miles
Repairs: The website Edmonds.com estimates that repairs on a three-year-old 2009
4Runner will be approximately $328 per year; this is based on driving 15,000 miles13
(6) Which would you recommend to Jenna, that she rent a car or use her own car for the trip? Explain
why.
12
13
Retrieved from www.fueleconomy.gov
Retrieved from www.edmunds.com/toyota/4runner/2006/tco.html?style=100614746
(7) Suppose Jenna will be taking a series of trips of varying lengths over the next month, and will be
using her own car for these trips. She wants to know the “cost per mile” for using her own car.
(a) What factors that affect the cost per mile might change over the course of a month?
(b) Write an expression for the cost per mile of using her car for these trips. Define any variables
you use. To think this through, you might start with these ideas:
- Cost per mile is (total costs) / (total miles).
- We don’t know the total miles since it varies in different months. So that will be a variable.
- To consider what all the costs are, remember that in the earlier question (5) you dealt with all
the costs for one particular trip.
(c) What factors that affect the cost per mile might Jenna be able to change over the course of a
year or longer?
Further Application:
Recall that whether Jenna rents a car or drives her own car, her employer will reimburse her the
same amount for the mileage driven using the rate set by the Internal Revenue Service. In 2015, that
rate was 57.5 cents/mile.
a) If Jenna drives her own car, will this reimbursement cover all of her extra costs? If so, how much
“profit” will she make?
b) What if Jenna rents a car – will this reimbursement cover all her extra costs? Does this depend
on other factors? If so, what?
1.7 The Fixer Upper
Specific Objectives
Students will understand that
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they can find formulas through the Internet and printed resources.
a variable can be used to represent an unknown.
using a formula requires knowing what each variable represents.
they must know the appropriate units for length, area, and volume.
units and dimensional analysis are helpful in working with formulas.
Students will be able to
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use formulas from geometry, use dimensional analysis, and perform calculations that involve
rates and measures to support financial decisions.
evaluate an expression.
Introduction to lesson
Geometric concepts of length, area, and volume are used in this lesson. Review the Resource page on
these topics.
Problem Situation: Home Improvements
Bob and Carol Mazursky have purchased a home and they want to make some improvements to it. In
the following problems, you will calculate the costs of those improvements. You will use a scale drawing
of the house and lot to assist you.
(1) Review the drawing of the house and lot (Figure 1 below).
(a) What does the scale of the drawing mean?
(b) Estimate and write the dimensions of the whole lot (all of their property).
- Use the dimensions to estimate the area of their lot. Include units with answers, of course.
- Use the dimensions to estimate the perimeter of their lot.
(c) Estimate the number of square feet in their house. (Assume it is a one-floor house.)
Figure 1: This scale drawing shows the rectangular lot, house, driveway, front yard, and back yard.
House
Front yard
Back yard
Driveway
20 ft
30 ft
(2) The Mazurskys are expecting their first child in several months and want to get the backyard
fertilized and reseeded before little Ted or Alice comes along. They found an ad for Gerry’s Green
Team lawn service (see below). Gerry came to their house and said that the job would take about
half a work day and would cost about $600. Is Gerry’s estimate consistent with his advertisement?
Gerry’s Green Team
Itemized Costs:
Grass seed
4 pounds per 1,000 sq. ft. @$1.25 per pound
Fertilizer
50 pounds per 12,000 sq. ft. @ $0.50 per pound
Labor
4 hours @ $45 per hour
advertisement
(3) The Mazurskys want to build a chain link fence around the backyard (the fence is already shown in
the diagram in Figure 1). The fence would have two gates on either side of the house. They decide to
do the work themselves. They need an inline post at least every 8 feet along the fence, a corner post
at each corner, and a corner post on each side of the gates. The cost for the materials is shown in
the advertisement below. The total cost will include 9.4% sales tax. Calculate the cost of the
materials required to fence in the backyard.
DO IT YOURSELF SPECIAL — Chain-Link Fence
Chain-link fence—$21 per linear yard
48-inch wide gate—$75 each
Inline posts—$12.50 each
Corner/gate posts—$20 each
HIGH Home Improvement—Your Fencing Specialist
(4) There is a brick grill in the backyard. Bob and Carol are going to make a concrete patio in the shape
of a semicircle next to the grill. (The patio is the shaded area marked in Figure 2.) The patio’s
concrete slab needs to be at least 2 inches thick. They will use 40-pound bags of premixed concrete.
Each 40-pound bag makes 0.30 cubic feet of concrete and costs $6.50. How much will the materials
cost, including the 9.4% tax?
Figure 2
1.8 Percentages in Many Forms
Specific Objectives
Students will understand that
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estimation is a valuable skill.
standard benchmarks can be used in estimation.
there are many strategies for estimating.
percentages are an important quantitative concept.
Students will be able to
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use a few standard benchmarks to estimate percentages (i.e., 1%, 10%, 25%, 33%, 50%, 66%,
75%).
estimate the percent one number is of another.
estimate the percent of a number, including situations involving percentages less than one.
calculate the percent one number is of another.
calculate the percent of a number, including situations involving percentages less than one.
Problem Situation: Estimations with Percentages
Strong estimation skills allow you to make quick calculations when it is inconvenient or unnecessary to
calculate exact results. You can also use estimation to check the results of a calculation. If the answer is
not close to your estimate, you know that you need to check your work. There are different types of
estimation.
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Educated guess: One type of estimation might be called an “educated guess” about something
that has not been measured exactly. In Lesson 1.2, you used estimations of the U.S debt. This
quantity cannot be measured exactly. Scientists can use good data and mathematical techniques
to estimate the debt, but it will always be an estimate.
Convenient estimation: Sometimes estimations are used when it is inconvenient or not
worthwhile to make an exact count. Imagine that you need to know how much paint to buy to
paint the baseboard trim in your house. (The baseboard trim is the piece of wood that follows
along the bottom of the walls.) You need to know the length of the baseboard. You could
measure the length of each wall to the nearest 1/8 inch and carefully subtract the width of halls
and doors. It would be much quicker and just as effective to measure to the nearest foot or half
foot. If you were cutting a piece of baseboard to go along the floor, however, you would want
an exact measurement.
Estimated calculation: This usually involves rounding numbers to make calculations simpler.
Lesson 1.8 focuses on estimating and calculating percentages. You will find in this course that
percentages are used in many contexts. One of the most important skills you will develop is
understanding and being comfortable working with percentages in a variety of situations.
In this course, you will make estimations and explain the strategies you used to generate estimations.
There is not one best strategy. It is important that you develop strategies that make sense to you. A
strategy is wrong only if it is mathematically incorrect (like saying that 25% is 1/2). In the following
section, you will practice your use of estimation strategies to answer the questions and calculate
percentages.
Use estimation to answer the following questions. Try to make your estimation calculations mentally.
Write down your work if you need to, but do not use a calculator. First, complete the problem yourself.
When you complete the problem, discuss your estimation strategy with your group. Your group should
find at least two different strategies for each problem.
1. Quick percent estimations
a) Describe two quick ways to find approximately 33% of 59 books.
b) Describe two quick ways to find approximately 25% of $98.50.
c) Describe two quick ways to find approximately 2% of 150 people.
d) Describe two quick ways to find approximately 0.5% of $14.95.
2. Quick fractions to percents
a) 5 out of 25 is _____%
b) 20 out of 80 is ____%
c) 3 out of 298 is ____%
d) 1 out of 1000 is ____%
3. You are shopping for a coat and find one that is on sale. The coat’s regular price is $87.99. What is
your estimate of the sale price based on each of the following discounts?
(a) 25% off
(b) 70% off
Estimations help you make calculations quickly in daily situations. This next problem shows how
estimates of percentages can be used to make comparisons among groups of different sizes.
4. A law enforcement officer reviews the following data from two precincts. She makes a quick
estimate to answer the following question: “If a violent incident occurs, in which precinct is it more
likely to involve a weapon?” Make an estimate to answer this question and explain your strategy.
Precinct
Number of Violent Incidents
Number of Violent Incidents
Involving a Weapon
1
25
5
2
122
18
5. You have a credit card that awards you a “cash back bonus.” This means that every time you use
your credit card to make a purchase, you earn back a percentage of the money you spend. Your card
gives you a bonus of 0.5%. Estimate your award on $462 in purchases.
From Estimation to Exact Calculation
Being able to calculate with percentages is also very important. In the situation in Question 1, an
estimate of the sale price will help you decide whether to buy the coat. However, the storeowner needs
to make an exact calculation to know how much to charge. In Question 2, an estimate helps the officer
get a sense of the situation, but if she is writing a report, she will want exact figures.
Calculate the exact answers for the situations in Questions 3 - 5. You may use a calculator. Show your
work.
6. If the coat’s regular price is $87.99, what is the exact sale price based on each of the following
discounts?
(a) 25% off
(b) 70% off
7. For each precinct, what is the exact percentage of incidents that involve a weapon? Round your
calculation to the nearest 1%.
8. Calculate the exact amount of your “cash back bonus” if your credit card awards a 0.5% bonus and
you charge $462 on your credit card.
1.9 The Credit Crunch
Specific Objectives
Students will understand that
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quantitative reasoning and math skills can be applied in various contexts.
creditworthiness affects credit card interest rates and the amount paid by the consumer.
reading quantitative information requires filtering out unimportant information.
Students will be able to
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recognize common mathematical concepts used in different contexts.
apply skills and concepts from previous lessons in new contexts.
identify a complete response to a prompt asking for connections between mathematical
concepts and a context.
write a statement about quantitative information in context.
write a spreadsheet formula.
Problem Situation: Understanding Credit Cards
When you use a credit card, you can pay off the entire balance at the end of the month and avoid paying
any interest. If you do not pay the full amount, you are borrowing money from the credit card company.
This is called credit card debt. Many people in the United States are concerned about the amount of
credit card debt for both individuals and for society in general. In this lesson, you will use skills and ideas
from previous lessons to think about some issues related to credit cards. You may want to refer back to
the previous lessons.
(1) According to the Federal Reserve System the total credit card debt carried by Americans as of March
2015 was 848.1 billion dollars.14 Write this amount in three other ways (words, number-word
combination, scientific notation, standard notation, etc.)
You will use the following information from a credit card disclosure for Questions 2 and 3.
Annual Percentage
Rate (APR) for
Purchases
14
15
0.00% introductory APR for 6 months from the date of account opening.
After that, your APR will be 10.99% to 23.99% based on your creditworthiness.
This APR will vary with the market based on the Prime Rate 15.
Retrieved from http://www.federalreserve.gov/releases/g19/current/
“Prime rate” is a base interest rate that banks charge their commercial customers.
(2) The Annual Percentage Rate varies with the market based on the Prime Rate. What is “Prime Rate”?
(3) APR stands for Annual Percentage Rate. It is the total interest rate for the entire year. However, we
normally make a credit card payment each month. The amount of interest paid each month is called
the Periodic Rate. Find the monthly Periodic Rate for an APR of 10.99%, rounded to two decimal
places.
(4) Creditworthiness is measured by a “credit score,” with a high credit score indicating good credit. In
the following questions, you will explore how your credit score can affect how much you have to pay
in order to borrow money. Juanita and Brian both have a credit card with the terms in the disclosure
form given above. They have both had their credit cards for more than 6 months.
(a) Juanita has good credit and gets the lowest interest rate possible for this card. She is not able to
pay off her balance each month, so she pays interest. Estimate how much interest Juanita would
pay in the month of January if her unpaid balance is $5000. Explain your estimation strategy.
(b) If Juanita maintains an average balance of $5000 every month for a year, estimate how much
interest she will pay in a year. Explain your estimation strategy.
(c) Brian has a very low credit score and has to pay the highest interest rate. He is not able
to pay off his balance each month, so he pays interest. Calculate how much interest he would
pay in the month of January if his balance is $5000.
(d) If Brian maintains an average balance of $5000 every month for a year, calculate how much
interest he will pay in a year.
You will use the following information from the disclosure for Question 5. A cash advance is when you
use your credit card to get cash instead of using it to make a purchase.
Annual Percentage
Rate (APR) for
Purchases
After that, your APR will be 10.99% to 23.99% based on your
creditworthiness. This APR will vary with the market based on the Prime Rate.
APR for Cash
Advances
28.99%. This APR will vary with the market based on the Prime Rate.
(5) Discuss each of the following statements. Decide if it is a reasonable statement.
(a) Jeff pays the highest interest rate for purchases. For a cash advance, he would pay $0.05 more
for each dollar he charges to his card.
(b) Lois pays the lowest interest rate for purchases. If she purchased a $400 TV using a cash
advance, she would pay about two-and-a-half times as much interest as she would if she used
the card as a regular purchase.
Brian used a spreadsheet to record his credit card charges for a month.
Brian entered the following formula in cell B7 to calculate his interest for these charges for one month.
=  0.2399 /12 *( B2  B3  B4  B5)
(6) Which of the following statements best explains what the expression means in terms of the context?
(i) Brian added his individual charges. Then he divided 0.2399 by 12. Then he multiplied the two
numbers.
(ii) Brian found the interest charge for the month by dividing 0.2399 by 12 and multiplying it by the
sum of Column B.
(iii) Brian found the periodic rate by dividing his APR of 0.2399 by 12 months. He then added the
individual charges to get the total amount charged to the credit card. He multiplied the periodic
rate by the total charges to find the interest charge for the month.
(7) Write two other formulas that Brian could have used to calculate his interest charge.
(8) Write a statement about one of these formulas that explains the mathematical expression in its
context.
(9) What are some things that might affect your credit score?
Review for Exam 1
Major Concepts
Large numbers
Writing principle
Scientific notation
Exponent rules
Numbers
Algebraic statements
Water Footprint
Dimensional Analysis
Cost of driving
Home improvements
Percentages – estimate and
calculate
Credit Cards